Notice that therefore any connection, even if not self-dual, is in some instanton sector, as its underlying bundle has some second Chern class, meaning that it can be obtained from shifting a self-dual connection. The self-dual connections are a convenient choice of “base point” in each instanton sector.

These must be such that there is t1<t2∈ℝt_1 \lt t_2 \in \mathbb{R} such that F∇(t<t1)=0F_\nabla(t \lt t_1) = 0 and F∇(T>t2)=0F_\nabla(T \gt t_2) = 0, hence these must be solutions interpolating between two flat connections ∇t1\nabla_{t_1} and ∇t2\nabla_{t_2}.

Since flat connections are the critical loci of SCSS_{CS} this says that a finite-action Yang-Mills instanton on Σ×ℝ\Sigma \times \mathbb{R} is a gradient flow trajectory between two Chern-Simons theory vacua .

Often this is interpreted as saying that “a Yang-Mills instanton describes the tunneling? between two Chern-Simons theoryvacua”.